fixed point of the Bellman operator

A fixed point is an input that a function leaves unchanged — apply the function and you get the same thing back. For the Bellman operator, the fixed point is the value function so self-consistent that one more backup changes nothing: every state's value already equals the reward-plus-discounted-successor it implies. That special value function is exactly what dynamic programming is hunting for.

The expectation operator has a unique fixed point, the value of its policy; the optimality operator has a unique fixed point, the optimal value function. The Bellman equations are precisely the statement that the value function equals its own backup. Because the operator is a contraction, the fixed point is unique and every starting guess converges to it — which is why value iteration and policy evaluation work, and where they stop.